One of the first kinds of differentiation matrices to be studied was for Chebyshev polynomials, i.e. \(T_0(x)=1\) , \(T_1(x) = x\)  and \(T_{k+1} = 2xT_k(x)-T_{k-1}(x)\) ; alternatively, \(T_k(x) = \cos (k\arccos (x))\)  on \(-1\le x \le 1\) . See for instance [1]} or (more briefly) Chapter 2 of [2]}. For a thorough and modern introduction with application to the Chebfun software project see [3]}. The derivative of \(T_k(x)\)  is explicitly given in terms of \(T_0, T_1,...,T_{k-1}\)  as a sum, in [1]} and as a Maple program in [2]}.
\(\frac{dT_k(x)}{dx}={\left\lbrace \begin{array}{ll}0 & k=0 \\k(\frac{1+(-1)^{k-1}}{2})T_0+2k\sum _{j=0}^{\lfloor \frac{k-1}{2}\rfloor } T_{k-1-2j}(x)& k\ge 1 \:.\\\end{array}\right.}\) 
